IN THIS LECTURE, YOU WILL LEARN:

Transcription

1 IN THIS LECTURE, YOU WILL LEARN: the closed economy Solow model how a country s standard of living depends on its saving and population growth rates how to use the Golden Rule to find the optimal saving rate and capital stock 0

2 Why growth matters Data on infant mortality rates: 20% in the poorest 1/5 of all countries 0.4% in the richest 1/5 In Pakistan, 85% of people live on less than $2/day. One-fourth of the poorest countries have had famines during the past 3 decades. Poverty is associated with oppression of women and minorities. Economic growth raises living standards and reduces poverty. 1

3 % of population living on $2 per day or less Income and poverty in the world selected countries, 2000 Madagascar India Nepal Bangladesh Kenya China Botswana Brazil Peru Thailand Mexico Chile Russian Federation S. Korea $0 $5,000 $10,000 $15,000 $20,000 Income per capita in dollars

4 Why growth matters Anything that affects the long-run rate of economic growth even by a tiny amount will have huge effects on living standards in the long run. annual growth rate of income per capita 25 years increase in standard of living after 50 years 100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% 4

5 Why growth matters If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher from , the average person would have earned $2,782 more during the decade. 5

6 The lessons of growth theory can make a positive difference in the lives of hundreds of millions of people. These lessons help us understand why poor countries are poor design policies that can help them grow learn how our own growth rate is affected by shocks and our government s policies 6

7 10.1 Measuring the Standard of Living Figure 10-1 U.S. GDP Since 1890 Aggregate U.S. output has increased by a factor of 42 since CHAPTER 7 of 26 8 The logarithmic scale on the vertical axis allows for the same proportional increase in a variable to be represented by the same distance. 7

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9 The Solow model due to Robert Solow, won Nobel Prize for contributions to the study of economic growth a major paradigm: widely used in policy making benchmark against which most recent growth theories are compared looks at the determinants of economic growth and the standard of living in the long run 9

10 How Solow model is different from the static production model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink 2. L is no longer fixed: population growth causes it to grow 3. the consumption function is simpler 10

11 How Solow model is different from the static production model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. cosmetic differences 11

12 The production function In aggregate terms: Y = F (K, L) Define: y = Y/L = output per worker k = K/L = capital per worker Assume constant returns to scale: zy = F (zk, zl ) for any z > 0 Pick z = 1/L. Then Y/L = F (K/L, 1) y = F (k, 1) y = f(k) where f(k) = F(k, 1) 12

13 The production function Output per worker, y f(k) 1 MPK = f(k +1) f(k) Note: this production function exhibits diminishing MPK. Capital per worker, k 13

14 The national income identity Y = C + I (remember, no G ) In per worker terms: y = c + i where c = C/L and i = I/L 14

15 The consumption function s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L Consumption function: c = (1 s)y (per worker) 15

16 Saving and investment saving (per worker) = y c = y (1 s)y = sy National income identity is y = c + i Rearrange to get: i = y c = sy (investment = saving, like in chap. 3!) Using the results above, i = sy = sf(k) 16

17 Output, consumption, and investment Output per worker, y f(k) y 1 c 1 sf(k) i 1 k 1 Capital per worker, k 17

18 Depreciation Depreciation per worker, k = the rate of depreciation = the fraction of the capital stock that wears out each period k 1 Capital per worker, k 18

19 Capital accumulation The basic idea: Investment increases the capital stock, depreciation reduces it. Change in capital stock = investment depreciation k = i k Since i = sf(k), this becomes: k = s f(k) k 19

20 The equation of motion for k k = s f(k) k The Solow model s central equation Determines behavior of capital per worker over time which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consumption per person: c = (1 s) f(k) 20

21 The steady state k = s f(k) k If investment is just enough to cover depreciation [sf(k) = k], then capital per worker will remain constant: k = 0. This occurs at one value of k, denoted k *, called the steady state capital stock. 21

22 The steady state Investment and depreciation k sf(k) k * Capital per worker, k 22

23 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) investment k depreciation k 1 k * Capital per worker, k 23

24 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k 1 k 2 k * Capital per worker, k 24

25 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) investment k depreciation k 2 k * Capital per worker, k 25

26 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k 2 k * Capital per worker, k 26

27 Moving toward the steady state Investment and depreciation k = sf(k) k k sf(k) k k k * 2 k 3 Capital per worker, k 27

28 Moving toward the steady state Investment and depreciation Summary: As long as k < k *, investment will exceed depreciation, and k will continue to grow toward k *. k = sf(k) k k sf(k) k 3 k * Capital per worker, k 28

29 NOW YOU TRY Approaching k* from above Draw the Solow model diagram, labeling the steady state k *. On the horizontal axis, pick a value greater than k * for the economy s initial capital stock. Label it k 1. Show what happens to k over time. Does k move toward the steady state or away from it? 29

30 A numerical example Production function (aggregate): Y F ( K, L) K L K L 1 / 2 1 / 2 To derive the per-worker production function, divide through by L: 1 / 2 1 / 2 1 / 2 Y K L K L L L Then substitute y = Y/L and k = K/L to get y f ( k ) k 1 / 2 30

31 A numerical example, cont. Suppose: s = 0.3 = 0.1 initial value of k =

32 Approaching the steady state: A numerical example Year k y c i k k 1 Assum

33 NOW YOU TRY Solve for the steady state Continue to assume s = 0.3, = 0.1, and y = k 1/2 Use the equation of motion k = s f(k) k to solve for the steady-state values of k, y, and c. 33

34 ANSWERS Solve for the steady state k 0 def. of steady state s f ( k *) k * eq'n of motion with k k * 0.1 k * using assumed values k * 3 k * k * Solve to get: k * 9 and y* k * 3 Finally, c * (1 s ) y *

35 An increase in the saving rate An increase in the saving rate raises investment causing k to grow toward a new steady state: Investment and depreciation k s 2 f(k) s 1 f(k) * k 1 * k 2 k 35

36 Prediction: Higher s higher k *. And since y = f(k), higher k * higher y *. Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run. 36

37 International evidence on investment rates and income per person Income per person in 2009 (log scale) 100,000 10,000 1, Investment as percentage of output (average )

38 The Golden Rule: Introduction Different values of s lead to different steady states. How do we know which is the best steady state? The best steady state has the highest possible consumption per person: c* = (1 s) f(k*). An increase in s leads to higher k* and y*, which raises c* reduces consumption s share of income (1 s), which lowers c*. So, how do we find the s and k* that maximize c*? 38

39 The Golden Rule capital stock * k gold the Golden Rule level of capital, the steady state value of k that maximizes consumption. To find it, first express c * in terms of k * : c * = y * i * = f(k * ) i * = f(k * ) k * In the steady state: i * = k * because k = 0. 39

40 The Golden Rule capital stock Then, graph f(k * ) and k *, look for the point where the gap between them is biggest. y f ( k ) * * gold gold steady state output and depreciation * k gold * c gold i k * * gold gold k * f(k * ) steady-state capital per worker, k * 40

41 The Golden Rule capital stock c * = f(k * ) k * is biggest where the slope of the production function equals the slope of the depreciation line: MPK = * c gold k * f(k * ) * k gold steady-state capital per worker, k * 41

42 The transition to the Golden Rule steady state The economy does NOT have a tendency to move toward the Golden Rule steady state. Achieving the Golden Rule requires that policymakers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule? 42

43 Starting with too much capital If k k * * gold then increasing c * requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. y c i t 0 time 43

44 Starting with too little capital * * If k kgold then increasing c * requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. y c i t 0 time 44

45 Population growth Assume the population and labor force grow at rate n (exogenous): L n L EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). Then L = nl = ,000 = 20, so L = 1,020 in year 2. 45

46 Break-even investment ( + n)k = break-even investment, the amount of investment necessary to keep k constant. Break-even investment includes: k to replace capital as it wears out nk to equip new workers with capital (Otherwise, k would fall as the existing capital stock is spread more thinly over a larger population of workers.) 46

47 The equation of motion for k With population growth, the equation of motion for k is: k = s f(k) ( + n) k actual investment break-even investment 47

48 The Solow model diagram Investment, break-even investment k = s f(k) ( +n)k ( + n )k sf(k) k * Capital per worker, k 48

49 The impact of population growth An increase in n causes an increase in breakeven investment, leading to a lower steady-state level of k. Investment, break-even investment ( +n 2 )k ( +n 1 )k sf(k) k 2 * k 1 * Capital per worker, k 49

50 Prediction: Higher n lower k*. And since y = f(k), lower k* lower y*. Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run. 50

51 International evidence on population growth and income per person Income per person in 2009 (log scale) 100,000 10,000 1, Population growth (percent per year, average )

52 The Golden Rule with population growth To find the Golden Rule capital stock, express c * in terms of k * : c * = y * i * = f(k * ) ( + n) k * c * is maximized when MPK = + n or equivalently, MPK = n In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate. 52

53 Alternative perspectives on population growth The Malthusian Model (1798) Predicts population growth will outstrip the Earth s ability to produce food, leading to the impoverishment of humanity. Since Malthus, world population has increased sixfold, yet living standards are higher than ever. Malthus neglected the effects of technological progress. 53

54 Alternative perspectives on population growth The Kremerian Model (1993) Posits that population growth contributes to economic growth. More people = more geniuses, scientists & engineers, so faster technological progress. Evidence, from very long historical periods: As world pop. growth rate increased, so did rate of growth in living standards Historically, regions with larger populations have enjoyed faster growth. 54

55 NEXT YOU WILL LEARN: how to incorporate technological progress in the Solow model about policies to promote growth about growth empirics: confronting the theory with facts 55

56 Examples of technological progress From 1950 to 2000, U.S. farm sector productivity nearly tripled. The real price of computer power has fallen an average of 30% per year over the past three decades. 2000: 361 million Internet users, 740 million cell phone users 2010: 2.0 billion Internet users, 3.8 billion cell phone users 2001: ipod capacity = 5gb, 1000 songs. Not capable of playing episodes of True Blood. 2011: ipod touch capacity = 64gb, 16,000 songs. Can play episodes of True Blood. 56

57 Technological progress in the Solow model A new variable: E = labor efficiency Assume: Technological progress is labor-augmenting: it increases labor efficiency at the exogenous rate g: g E E 57

58 Technological progress in the Solow model We now write the production function as: Y F ( K, L E ) where L E = the number of effective workers. Increases in labor efficiency have the same effect on output as increases in the labor force. 58

59 Technological progress in the Solow model Notation: y = Y/LE = output per effective worker k = K/LE = capital per effective worker Production function per effective worker: y = f(k) Saving and investment per effective worker: sy = sf(k) 59

60 Technological progress in the Solow model ( + n + g)k = break-even investment: the amount of investment necessary to keep k constant. Consists of: k to replace depreciating capital n k to provide capital for new workers g k to provide capital for the new effective workers created by technological progress 60

61 Technological progress in the Solow model Investment, break-even investment k = s f(k) ( +n +g)k ( +n +g )k sf(k) k * Capital per worker, k 61

62 Steady-state growth rates in the Solow model with tech. progress Variable Capital per effective worker Output per effective worker Output per worker Symbol k = K / (L E ) y = Y / (L E ) (Y/ L) = y E Steady-state growth rate 0 0 g Total output Y = y E L n + g 62

63 The Golden Rule with technological progress To find the Golden Rule capital stock, express c * in terms of k * : c * = y * i * = f (k * ) ( +n +g)k * c * is maximized when MPK = + n + g or equivalently, MPK = n + g In the Golden Rule steady state, the marginal product of capital net of depreciation equals the pop. growth rate plus the rate of tech progress. 63

64 Growth empirics: Balanced growth Solow model s steady state exhibits balanced growth many variables grow at the same rate. Solow model predicts Y/L and K/L grow at the same rate (g), so K/Y should be constant. This is true in the real world. Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant. Also true in the real world. 64

65 Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. If true, then the income gap between rich & poor countries would shrink over time, causing living standards to converge. In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails? 65

66 Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. No, because other things aren t equal: In samples of countries with similar savings & pop. growth rates, income gaps shrink about 2% per year. In larger samples, after controlling for differences in saving, pop. growth, and human capital, incomes converge by about 2% per year. 66

67 Growth empirics: Convergence What the Solow model really predicts is conditional convergence countries converge to their own steady states, which are determined by saving, population growth, and education. This prediction comes true in the real world. 67

68 Growth empirics: Factor accumulation vs. production efficiency Differences in income per capita among countries can be due to differences in: 1. capital physical or human per worker 2. the efficiency of production (the height of the production function) Studies: Both factors are important. The two factors are correlated: countries with higher physical or human capital per worker also tend to have higher production efficiency. 68

69 Growth empirics: Factor accumulation vs. production efficiency Possible explanations for the correlation between capital per worker and production efficiency: Production efficiency encourages capital accumulation. Capital accumulation has externalities that raise efficiency. A third, unknown variable causes capital accumulation and efficiency to be higher in some countries than others. 69

70 Policy issues Are we saving enough? Too much? What policies might change the saving rate? How should we allocate our investment between privately owned physical capital, public infrastructure, and human capital? How do a country s institutions affect production efficiency and capital accumulation? What policies might encourage faster technological progress? 72

71 Policy issues: Evaluating the rate of saving Use the Golden Rule to determine whether the U.S. saving rate and capital stock are too high, too low, or about right. If (MPK ) > (n + g ), U.S. economy is below the Golden Rule steady state and should increase s. If (MPK ) < (n + g ), U.S. economy is above the Golden Rule steady state and should reduce s. 73

72 Policy issues: Evaluating the rate of saving To estimate (MPK ), use three facts about the U.S. economy: 1. k = 2.5 y 2. k = 0.1 y 3. MPK k = 0.3 y To determine, divide 2 by 1: k k 0.1 y 2.5y

73 Policy issues: Evaluating the rate of saving 1. k = 2.5 y 2. k = 0.1 y 3. MPK k = 0.3 y To determine MPK, divide 3 by 1: MPK k 0.3y k 2.5y 0.3 MPK Hence, MPK = =

74 Policy issues: Evaluating the rate of saving From the last slide: MPK = 0.08 U.S. real GDP grows an average of 3% per year, so n + g = 0.03 Thus, MPK = 0.08 > 0.03 = n + g Conclusion: The U.S. is below the Golden Rule steady state: Increasing the U.S. saving rate would increase consumption per capita in the long run. 77

75 Policy issues: How to increase the saving rate Reduce the government budget deficit (or increase the budget surplus). Increase incentives for private saving: Reduce capital gains tax, corporate income tax, estate tax, as they discourage saving. Replace federal income tax with a consumption tax. Expand tax incentives for IRAs (individual retirement accounts) and other retirement savings accounts. 78

76 Policy issues: Allocating the economy s investment In the Solow model, there s one type of capital. In the real world, there are many types, which we can divide into three categories: private capital stock public infrastructure human capital: the knowledge and skills that workers acquire through education How should we allocate investment among these types? 79

77 Policy issues: Allocating the economy s investment Two viewpoints: 1. Equalize tax treatment of all types of capital in all industries, then let the market allocate investment to the type with the highest marginal product. 2. Industrial policy: Govt should actively encourage investment in capital of certain types or in certain industries, because they may have positive externalities that private investors don t consider. 80

78 Possible problems with industrial policy The govt may not have the ability to pick winners (choose industries with the highest return to capital or biggest externalities). Politics (e.g., campaign contributions) rather than economics may influence which industries get preferential treatment. 81

79 Policy issues: Establishing the right institutions Creating the right institutions is important for ensuring that resources are allocated to their best use. Examples: Legal institutions, to protect property rights. Capital markets, to help financial capital flow to the best investment projects. A corruption-free government, to promote competition, enforce contracts, etc. 82

80 Policy issues: Encouraging tech. progress Patent laws: encourage innovation by granting temporary monopolies to inventors of new products. Tax incentives for R&D Grants to fund basic research at universities Industrial policy: encourages specific industries that are key for rapid tech. progress (subject to the preceding concerns). 83

81 CASE STUDY: The worldwide slowdown Canada France Germany Italy Japan U.K. U.S. Growth in output per person (percent per year)

82 Possible explanations for the slowdown Measurement problems: Productivity increases not fully measured. But: Why would measurement problems be worse after 1972 than before? Oil prices: Oil shocks occurred about when productivity slowdown began. But: Then why didn t productivity speed up when oil prices fell in the mid-1980s? 85

83 Possible explanations for the slowdown Worker quality: 1970s large influx of new entrants into labor force (baby boomers, women). New workers tend to be less productive than experienced workers. The depletion of ideas: Perhaps the slow growth of is normal, and the rapid growth during is the anomaly. 86

84 Which of these suspects is the culprit? All of them are plausible, but it s difficult to prove that any one of them is guilty. 87

85 CASE STUDY: The worldwide slowdown Canada France Germany Italy Japan U.K. U.S. Growth in output per person (percent per year)

86 CASE STUDY: The worldwide slowdown The computer revolution and Internet began to affect aggregate productivity in mid-1990s continuing into the mid 2000s. This period was then offset by the financial crisis and deep recession of

87 L E C T U R E S U M M A R Y 1. The Solow growth model shows that, in the long run, a country s standard of living depends: positively on its saving rate negatively on its population growth rate 2. An increase in the saving rate leads to: higher output in the long run faster growth temporarily but not faster steady-state growth 90

88 L E C T U R E S U M M A R Y 3. If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off. If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation. 91